where \(n\)  is the number of samples from the posterior under
model \(M_{}\) .
[1]} used this approach to show that a mixture model that
accommodates among-site heterogeneity in amino acid frequencies is a better
predictor of animal sequence alignments than standard amino-acid models.
This corroborated the findings of [2]} based on path-sampling
estimates of marginal likelihoods.
[3]} introduced a leave-one-out cross-validation approach to
phylogenetics called the conditional predictive ordinates (CPO) method
[4]}, [5]}, [6]}.
This method leaves one site out of the alignment to serve as the testing data
to estimate \(p(D ^{\prime } \,|\, M_{}, D)\) , which is equal to the
posterior harmonic mean of the site likelihood [6]}.
Summing the log of this value across all sites yields what is called the log
pseudomarginal likelihood (LPML).
[3]} compared the estimated LPML to stepping-stone estimates of
the marginal likelihood for selecting among models that differed in how they
partitioned sites across a concatenated alignment of four genes from algae.
The LPML favored a 12-subset model (partitioned by gene and codon position)
as opposed to the 3-subset model (partitioned by codon) preferred by
marginal likelihoods.
This difference could reflect the lesser penalty against additional parameters
imposed by the weight of the posterior
(Equation REF ) versus the prior
(Equation REF ).
